To better estimate the rock joint shear strength, accurately determining the rock joint roughness coefficient (JRC) is the first step faced by researchers and engineers. However, there are incomplete, imprecise, and indeterminate problems during the process of calculating the JRC. This paper proposed to investigate the indeterminate information of rock joint roughness through a neutrosophic number approach and, based on this information, reported a method to capture the incomplete, uncertain, and imprecise information of the JRC in uncertain environments. The uncertainties in the JRC determination were investigated by the regression correlations based on commonly used statistical parameters, which demonstrated the drawbacks of traditional JRC regression correlations in handling the indeterminate information of the JRC. Moreover, the commonly used statistical parameters cannot reflect the roughness contribution differences of the asperities with various scales, which induces additional indeterminate information. A method based on the neutrosophic number (NN) and spectral analysis was proposed to capture the indeterminate information of the JRC. The proposed method was then applied to determine the JRC values for sandstone joint samples collected from a rock landslide. The comparison between the JRC results obtained by the proposed method and experimental results validated the effectiveness of the NN. Additionally, comparisons made between the spectral analysis and common statistical parameters based on the NN also demonstrated the advantage of spectral analysis. Thus, the NN and spectral analysis combined can effectively handle the indeterminate information in the rock joint roughness.
Many major foundation projects have been constructed in complex geological conditions, and numerous high rock slopes are formed. The rock joints in the rock slopes are affected by the geological forces inside and outside the earth, making the slope easy to slip along the controlled joints [
The joint roughness coefficient (JRC) proposed by Muralha et al. [
Considering the contribution of different frequency components of the rock joint surface to the roughness are generally different, Wang et al. [
The NN and other neutrosophic theories such as neutrosophic statistics, neutrosophic probability, and neutrosophic distribution are major branches of the neutrosophic theory, which deals with indeterminate data and indeterminate inference methods that contain degrees of indeterminacy as well [
The structure of this paper is listed as follows. In Section 2, the basic concepts for neutrosophic number functions and spectral analysis are presented. Then, in Section 3, the uncertainties in joint roughness coefficient determination are first discussed. A new method to calculate the JRC based on neutrosophic number functions and spectral analysis is then proposed. In Section 4, the comparisons between the new approach and commonly used statistical parameters to determine the JRC are carried out based on experimental results of the rock joints collected from an actual rock landslides area. Finally, the conclusion is presented in Section 5.
Generally, a NN
where
For example, let us assume that a NN is
Generally, the rock joints in engineering practice consist of various scales of asperities. The asperities with small inclinations but high amplitudes are low-frequency components, while the asperities with big inclinations but low amplitudes are high-frequency components. Due to the different frequencies, the various scales of asperity components have different influences on the rock joint shear behavior. To quantitively describe the contribution of asperities with various frequencies on the rock joint roughness, Wang et al. [
Although the rock joint morphology in engineering practice is so complex that difficult to be described by clear math equations, the spectral characteristics of the profile are readily analyzed by the power spectral density (PSD) [
To simplify the calculation, the least-square fitting line of a rock joint profile is first aligned to be horizontal, and the average straight line of the profile is shifted to coincide with the coordinate axis
where
The average power and Fourier transform of the aligned profile
where
According to the Wiener–Khintchine theorem [
where PSD(
As indicated by
As the rock joint profile data collected in the engineering practice is discrete under a certain sampling interval, the discrete form of the PSD is presented as follows to facilitate practical applications.
where
Among the various quantitative JRC determination methods, the statistical and fractal methods are widely adopted by researchers and engineers. However, the joint profile in engineering practice is self-affine; different users may get contradictory results with the fractal method [
Researchers have been working on statistical parameters to determine the JRC quantitatively, and many regression correlations with high correlation coefficients have been proposed. However, existing regression correlations can only address the determinate information of JRC but cannot express and handle the indeterminate information. Therefore, there may exist deviations in the JRC calculation results for rock joints. To demonstrate the deviations arise from ignoring the incomplete, uncertain, and imprecise information in the JRC, two JRC regression relations based on the widely used statistical parameters
The ten standard profiles [
Statistical parameters | Formulations | Notations |
---|---|---|
Profile ID | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|
JRC_{true} | 0.4 | 2.8 | 5.8 | 6.7 | 9.5 | 10.8 | 12.8 | 14.5 | 16.7 | 18.7 | |
0.075 | 0.169 | 0.149 | 0.227 | 0.206 | 0.210 | 0.266 | 0.285 | 0.332 | 0.439 | ||
0.001 | 0.005 | 0.004 | 0.008 | 0.007 | 0.007 | 0.011 | 0.013 | 0.017 | 0.031 | ||
JRC_{Z2} | 1.3 | 4.9 | 4.0 | 8.1 | 6.9 | 7.1 | 10.6 | 11.9 | 15.4 | 24.7 | |
0.1 | 5.3 | 4.2 | 8.4 | 7.3 | 7.5 | 10.6 | 11.6 | 13.8 | 20.2 | ||
Deviations | JRC_{Z2} | 225.0% | 75.0% | −31.0% | 20.9% | −27.4% | −34.3% | −17.2% | −17.9% | −7.8% | 32.1% |
JRC_{SF} | −75.0% | 89.3% | −27.6% | 25.4% | −23.2% | −30.6% | −17.2% | −20.0% | −17.4% | 8.0% |
According to the spectral information presented by the rock joint profiles, Wang et al. [
where
In the study [
As suggested by the authors [
where
According to
The
Five well-matched sandstone joint samples were collected from the Majiagou landslide. The Majiagou landslide is in Guizhou Town, Zigui County, Yichang City, Hubei Province, China. It is located at the foot of Woniu Mountain on the left bank of the Yangtze River, on the left bank of the Zhaxi River, a tributary of the Yangtze River, and 2.1 km from the mouth of the Yangtze River. The bedrock stratum in the landslide area is the Upper Jurassic Suining Formation (J_{3S}), which belongs to the middle of the Guizhou Group. The lithology is mainly gray-white feldspar quartz sandstone and fine sandstone, with purple-red silty mudstone and mudstone. The sandstone joint samples collected in this paper are mainly gray-white feldspar quartz sandstone joints.
These collected five well-matched sandstone joint samples were cut into standard samples with a length and width of 10 cm and height of 5 cm. Then, a laser scanner was used to scan the surface of these samples with an accuracy of ±35 μm and a sampling interval of 0.2 mm. A photograph of the laser scanner and data acquisition system can be seen in
The 3D surfaces of the collected five sandstone joints were also constructed with the same sampling interval (
Sample IDs | MJ1-5 | MJ1-6 | MJ1-7 | MJ1-8 | MJ1-9 | |
---|---|---|---|---|---|---|
Normal stress (MPa) | 1 | 2.5 | 2 | 1.5 | 0.5 | |
Shear strength (MPa) | 1.035 | 2.172 | 1.527 | 1.341 | 0.415 | |
JRC_{back-calculated} | 11.5 | 11.2 | 7.8 | 10.1 | 6.6 | |
JRC_{predicted} | [9.0, 13.8] | [8.3, 13.1] | [5.9, 10.7] | [8.6, 13.4] | [4.3, 9.1] | |
[5.3, 14.3] | [4.2, 13.2] | [1.6, 10.6] | [4.8, 13.8] | [1.4, 10.4] | ||
[6.1, 14.1] | [4.8, 12.8] | [3.1, 11.1] | [6.0, 14.0] | [3.4, 11.4] | ||
[8.4, 15.4] | [7.7, 14.7] | [6.0, 13.0] | [8.6, 15.6] | [5.1, 12.1] | ||
[7.9, 15.9] | [7.9, 15.9] | [5.2, 13.2] | [8.3, 16.3] | [3.9, 11.9] | ||
[10.2, 17.7] | [11.8, 19.3] | [6.9, 14.4] | [10.2, 17.7] | [7.2, 14.7] | ||
[7.8, 15.8] | [6.0, 14.0] | [3.8, 11.8] | [7.8, 15.8] | [3.7, 11.7] | ||
[6.1, 15.1] | [4.1, 13.1] | [1.8, 10.8] | [6.1, 15.1] | [1.9, 10.9] | ||
[6.4, 14.9] | [4.4, 12.9] | [2.2, 10.7] | [6.4, 14.9] | [2.3, 10.8] |
The roughness parameter
For comparison purposes, the commonly used statistical parameters,
According to the derived statistical parameter-based NN functions,
Generally, the JRC values have good correlations with commonly used statistical parameters. However, the indeterminate information of the JRC leads to variabilities in the predicted results of JRC regression correlations. Researchers usually adopt the mean trend of correlations to predict JRC values, while this subjected approach may result in biased results. Here, the mean trend correlations of the above-mentioned statistical parameters and the proposed
Incomplete, imprecise, and indeterminate problems are generally encountered for the complex surface of rock joints. The NNs are preferred compared to fuzzy, rough, grey sets due to efficiency, flexibility, and easiness for expressing determinate and/or indeterminate information. Additionally, various scales of asperities are displayed on joint surfaces. The spectral analysis was adopted to simultaneously capture the contributions of the high and low-frequency asperity components in determining joint roughness. Through the combination of the NNs and spectral analysis, this paper proposed a new method to determine the JRC for rock joints accurately. The main conclusions are drawn as follows.
The JRC regression correlation based on commonly used statistical parameters could not handle the indeterminate information of the JRC. As a result, there were still significant deviations in calculated JRC values, although the JRC had a good correlation with the commonly used statistical parameters. Particularly, the absolute deviations for the calculated JRC results based on the
To overcome the limitations of the traditional JRC determination approaches, NN functions based on the spectral roughness parameter were derived based on 112 rock joint profiles collected from the literature. Then, the derived NN function was applied to determine the JRC values of 5 well-matched sandstone joint samples collected from the Majiagou rock landslide area. The comparison between the JRC results obtained from the proposed method and experimental results validated the effectiveness of the spectral analysis based NN functions. Additionally, comparisons made between the spectral analysis and common statistical parameters based on NN also demonstrated the advantage of spectral analysis. The combination of the NNs and spectral analysis can effectively address the indeterminate information that exists in the joint roughness and the contribution differences of various scales of asperities on the joint roughness.
In addition to the NN, the neutrosophic theory contains many other methods, such as neutrosophic sets, neutrosophic interval statistical numbers, and neutrosophic interval functions. They can also address the indeterminate information in the JRC determination process. In future work, we will further develop and investigate JRC determination approaches through other neutrosophic theories and compare the differences and efficiencies between different neutrosophic approaches.